Subgroups — C₃×#10592436 #13162774

Order 1728.

836 conjugacy classes of subgroups.

The subgroup lattice has 358 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃×#64139	C₂	1	1	yes	maximal, characteristic
2	C₃×#64135	C₂	1	1	yes	maximal, characteristic, complemented
3	C₃³×C₂²⋊C₈	C₂	1	1	yes	maximal, characteristic, complemented
4	C₃×#63648	C₂	1	1	yes	maximal, characteristic, complemented
5	C₃×#63648	C₂	1	1	yes	maximal, characteristic, complemented
6	C₃×#63645	C₂	1	1	yes	maximal, characteristic, complemented
7	C₃×#63644	C₂	1	1	yes	maximal, characteristic, complemented
8	#10592436	C₃	1	1	yes	maximal, characteristic, direct product factor
9	C₃×#4737	—	3	4	no	maximal
10	C₂²×C₃³×C₄	C₂²	1	1	yes	characteristic
11	C₂×C₃×#638	—	2	1	no	—
12	C₂×C₃×C₄⋊(C₃⋊D₃)	C₂²	1	1	yes	characteristic
13	C₂×C₃³×C₈	C₂²	1	1	yes	characteristic
14	C₂×C₃³×C₈	C₂²	1	1	yes	characteristic
15	C₃×#3757	—	2	1	no	—
16	C₃×#3752	—	2	1	no	—
17	C₃×C₄⋊(C₃⋊Dic₃)	C₂²	1	1	yes	characteristic
18	C₃×C₄⋊(C₃⋊Dic₃)	C₂²	1	1	yes	characteristic
19	C₃×C₄⋊(C₃⋊Dic₃)	C₂²	1	1	yes	characteristic
20	C₃×C₄×C₃⋊Dic₃	—	2	1	no	—
21	#64139	C₂×C₃	1	1	yes	characteristic
22	#64135	C₂×C₃	1	1	yes	characteristic, complemented
23	C₃×#930	—	3	4	no	—
24	C₃×#926	—	3	4	no	—
25	C₃²×C₂²⋊C₈	—	2	4	no	—
26	C₃²×C₂²⋊C₈	D₃	1	4	yes	complemented
27	C₃²×C₂²⋊C₈	C₂×C₃	1	1	yes	characteristic, complemented
28	#63648	C₂×C₃	1	1	yes	characteristic, complemented
29	#63648	C₂×C₃	1	1	yes	characteristic, complemented
30	#63645	C₂×C₃	1	1	yes	characteristic, complemented
31	#63644	C₂×C₃	1	1	yes	characteristic, complemented
32	C₃×#821	—	3	4	no	—
33	C₃×#821	—	3	4	no	—
34	C₃×#818	—	3	4	no	—
35	C₃×#817	—	3	4	no	—
36	C₂³×C₃³	D₄	1	1	yes	characteristic
37	C₂²×C₃×C₃⋊D₃	—	2	1	no	—
38	C₂×C₃³×C₄	—	2	1	no	—
39	C₂×C₃³×C₄	—	2	1	no	—
40	C₂×C₃³×C₄	D₄	1	1	yes	characteristic
41	C₂×C₃³×C₄	C₂³	1	1	yes	characteristic
42	C₃×#638	—	4	2	no	—
43	C₂×C₃×C₃⋊Dic₃	—	2	1	no	—
44	C₂×C₃×C₃⋊Dic₃	—	2	1	no	—
45	C₂×C₃×C₃⋊Dic₃	—	2	1	no	—
46	C₃×C₄⋊(C₃⋊D₃)	—	2	1	no	—
47	C₃×C₄⋊(C₃⋊D₃)	—	2	1	no	—
48	C₃³×C₈	—	2	1	no	—
49	C₃³×C₈	—	2	1	no	—
50	C₃×#482	—	9	1	no	—
51	#4737	—	3	4	no	—
52	C₂²×C₃²×C₄	—	2	4	no	—
53	C₂²×C₃²×C₄	C₂×D₃	1	4	yes	—
54	C₂²×C₃²×C₄	C₂²×C₃	1	1	yes	characteristic
55	C₂×#638	—	2	1	no	—
56	C₂×C₄⋊(C₃⋊D₃)	C₂²×C₃	1	1	yes	characteristic
57	C₂×C₃×#67	—	6	4	no	—
58	C₂×C₃×D₁₂	—	3	4	no	—
59	C₂×C₃²×C₈	—	2	4	no	—
60	C₂×C₃²×C₈	—	2	4	no	—
61	C₂×C₃²×C₈	C₂×D₃	1	4	yes	—
62	C₂×C₃²×C₈	C₂×D₃	1	4	yes	—
63	C₂×C₃²×C₈	C₂²×C₃	1	1	yes	characteristic
64	C₂×C₃²×C₈	C₂²×C₃	1	1	yes	characteristic
65	#3757	—	2	1	no	—
66	#3752	—	2	1	no	—
67	C₄⋊(C₃⋊Dic₃)	C₂²×C₃	1	1	yes	characteristic
68	C₄⋊(C₃⋊Dic₃)	C₂²×C₃	1	1	yes	characteristic
69	C₄⋊(C₃⋊Dic₃)	C₂²×C₃	1	1	yes	characteristic
70	C₄×C₃⋊Dic₃	—	2	1	no	—
71	C₃×#217	—	6	4	no	—
72	C₃×(C₂×D₃)⋊C₄	—	6	4	no	—
73	C₃×C₄⋊Dic₃	—	3	4	no	—
74	C₃×C₄⋊Dic₃	—	3	4	no	—
75	C₃×C₄⋊Dic₃	—	3	4	no	—
76	C₃×C₄×Dic₃	—	6	4	no	—
77	C₂²×C₃³	—	2	1	no	—
78	C₂²×C₃³	—	2	1	no	—
79	C₂²×C₃³	—	2	1	no	—
80	C₂²×C₃³	C₂×D₄	1	1	yes	characteristic
81	C₂×C₃×C₃⋊D₃	—	4	1	no	—
82	C₂×C₃×C₃⋊D₃	—	4	1	no	—
83	C₂×C₃×C₃⋊D₃	—	4	1	no	—
84	C₃³×C₄	—	2	1	no	—
85	C₃³×C₄	#41	1	1	yes	characteristic
86	C₃³×C₄	#41	1	1	yes	characteristic
87	C₃×C₃⋊Dic₃	—	4	1	no	—
88	C₃×C₃⋊Dic₃	—	4	1	no	—
89	C₃×C₃⋊Dic₃	—	4	1	no	—
90	C₃×#121	—	9	1	no	—
91	C₃×#117	—	9	1	no	—
92	#930	—	3	4	no	—
93	#926	—	3	4	no	—
94	C₃×#107	—	9	1	no	—
95	C₃×#106	—	9	1	no	—
96	C₃×#102	—	9	1	no	—
97	C₃×#102	—	9	1	no	—
98	C₃×C₂²⋊C₈	—	2	4	no	—
99	C₃×C₂²⋊C₈	C₃×D₃	1	4	yes	complemented
100	C₃×C₂²⋊C₈	C₃⋊D₃	1	1	yes	characteristic, complemented
101	#821	—	3	4	no	—
102	#821	—	3	4	no	—
103	#818	—	3	4	no	—
104	#817	—	3	4	no	—
105	C₂³×C₃²	—	2	4	no	—
106	C₂³×C₃²	D₁₂	1	4	yes	—
107	C₂³×C₃²	C₃×D₄	1	1	yes	characteristic
108	C₂²×C₃⋊D₃	—	2	1	no	—
109	C₂²×C₃×D₃	—	6	4	no	—
110	C₂×C₃²×C₄	—	2	1	no	—
111	C₂×C₃²×C₄	—	2	1	no	—
112	C₂×C₃²×C₄	—	2	4	no	—
113	C₂×C₃²×C₄	—	2	4	no	—
114	C₂×C₃²×C₄	—	2	4	no	—
115	C₂×C₃²×C₄	—	2	4	no	—
116	C₂×C₃²×C₄	—	4	4	no	—
117	C₂×C₃²×C₄	—	4	4	no	—
118	C₂×C₃²×C₄	D₁₂	1	4	yes	—
119	C₂×C₃²×C₄	C₃×D₄	1	1	yes	characteristic
120	C₂×C₃²×C₄	C₂²×D₃	1	4	yes	—
121	C₂×C₃²×C₄	C₂³×C₃	1	1	yes	characteristic
122	#638	—	4	2	no	—
123	C₂×C₃⋊Dic₃	—	2	1	no	—
124	C₂×C₃⋊Dic₃	—	2	1	no	—
125	C₂×C₃⋊Dic₃	—	2	1	no	—
126	C₄⋊(C₃⋊D₃)	—	2	1	no	—
127	C₄⋊(C₃⋊D₃)	—	2	1	no	—
128	C₃×#67	—	12	8	no	—
129	C₂×C₃×Dic₃	—	6	4	no	—
130	C₂×C₃×Dic₃	—	6	4	no	—
131	C₂×C₃×Dic₃	—	6	4	no	—
132	C₃×D₁₂	—	6	4	no	—
133	C₃×D₁₂	—	6	4	no	—
134	C₃²×C₈	—	2	1	no	—
135	C₃²×C₈	—	2	1	no	—
136	C₃²×C₈	—	2	4	no	—
137	C₃²×C₈	—	2	4	no	—
138	C₃²×C₈	—	4	4	no	—
139	C₃²×C₈	—	4	4	no	—
140	#482	—	9	1	no	Sylow
141	C₂×C₃³	—	4	1	no	—
142	C₂×C₃³	#123	1	1	yes	characteristic
143	C₂×C₃³	#135	1	1	yes	characteristic
144	C₂×C₃³	AGL(1,ℤ/8)	1	1	yes	characteristic
145	C₃×C₃⋊D₃	—	8	1	no	—
146	C₂×C₃×D₄	—	9	1	no	—
147	C₂×C₃×D₄	—	18	1	no	—
148	C₂²×C₃×C₄	—	2	4	no	—
149	C₂²×C₃×C₄	C₂×C₃×D₃	1	4	yes	—
150	C₂²×C₃×C₄	C₂×C₃⋊D₃	1	1	yes	characteristic
151	C₂×#67	—	6	4	no	—
152	C₂×D₁₂	—	3	4	no	—
153	C₂×C₃×C₈	—	2	4	no	—
154	C₂×C₃×C₈	—	2	4	no	—
155	C₂×C₃×C₈	C₂×C₃×D₃	1	4	yes	—
156	C₂×C₃×C₈	C₂×C₃×D₃	1	4	yes	—
157	C₂×C₃×C₈	C₂×C₃⋊D₃	1	1	yes	characteristic
158	C₂×C₃×C₈	C₂×C₃⋊D₃	1	1	yes	characteristic
159	C₃×C₄⋊C₄	—	9	1	no	—
160	C₃×C₄⋊C₄	—	9	1	no	—
161	C₃×C₄⋊C₄	—	9	1	no	—
162	C₃×C₂²⋊C₄	—	18	1	no	—
163	C₃×C₂²⋊C₄	—	18	1	no	—
164	C₃×C₄²	—	18	1	no	—
165	#217	—	6	4	no	—
166	(C₂×D₃)⋊C₄	—	6	4	no	—
167	C₄⋊Dic₃	—	3	4	no	—
168	C₄⋊Dic₃	—	3	4	no	—
169	C₄⋊Dic₃	—	3	4	no	—
170	C₄×Dic₃	—	6	4	no	—
171	C₂²×C₃²	—	2	8	no	—
172	C₂²×C₃²	—	2	4	no	—
173	C₂²×C₃²	—	2	1	no	—
174	C₂²×C₃²	—	2	1	no	—
175	C₂²×C₃²	—	2	1	no	—
176	C₂²×C₃²	—	2	4	no	—
177	C₂²×C₃²	—	2	4	no	—
178	C₂²×C₃²	—	2	4	no	—
179	C₂²×C₃²	—	4	4	no	—
180	C₂²×C₃²	—	4	4	no	—
181	C₂²×C₃²	C₂×D₁₂	1	4	yes	—
182	C₂²×C₃²	C₂×C₃×D₄	1	1	yes	characteristic
183	C₂×C₃⋊D₃	—	4	1	no	—
184	C₂×C₃⋊D₃	—	4	1	no	—
185	C₂×C₃⋊D₃	—	4	1	no	—
186	C₂×C₃×D₃	—	12	4	no	—
187	C₂×C₃×D₃	—	12	4	no	—
188	C₂×C₃×D₃	—	12	4	no	—
189	C₃²×C₄	—	2	1	no	—
190	C₃²×C₄	—	2	4	no	—
191	C₃²×C₄	—	2	4	no	—
192	C₃²×C₄	—	2	4	no	—
193	C₃²×C₄	—	4	4	no	—
194	C₃²×C₄	#237	1	4	yes	—
195	C₃²×C₄	#237	1	4	yes	—
196	C₃²×C₄	C₃×#41	1	1	yes	characteristic
197	C₃²×C₄	C₃×#41	1	1	yes	characteristic
198	C₃⋊Dic₃	—	4	1	no	—
199	C₃⋊Dic₃	—	4	1	no	—
200	C₃⋊Dic₃	—	4	1	no	—
201	C₃×Dic₃	—	12	4	no	—
202	C₃×Dic₃	—	12	4	no	—
203	C₃×Dic₃	—	12	4	no	—
204	#121	—	9	1	no	—
205	#117	—	9	1	no	—
206	#107	—	9	1	no	—
207	#106	—	9	1	no	—
208	#102	—	9	1	no	—
209	#102	—	9	1	no	—
210	C₂²⋊C₈	C₃×C₃⋊D₃	1	1	yes	characteristic, complemented
211	C₃³	#482	1	1	yes	characteristic, complemented, Sylow
212	C₂³×C₃	—	2	4	no	—
213	C₂³×C₃	—	18	1	no	—
214	C₂³×C₃	C₃×D₁₂	1	4	yes	—
215	C₂³×C₃	C₄⋊(C₃⋊D₃)	1	1	yes	characteristic
216	C₂²×D₃	—	6	4	no	—
217	C₃×D₄	—	18	1	no	—
218	C₃×D₄	—	18	1	no	—
219	C₃×D₄	—	36	2	no	—
220	C₂×C₃×C₄	—	2	4	no	—
221	C₂×C₃×C₄	—	2	4	no	—
222	C₂×C₃×C₄	—	2	1	no	—
223	C₂×C₃×C₄	—	2	1	no	—
224	C₂×C₃×C₄	—	2	4	no	—
225	C₂×C₃×C₄	—	2	4	no	—
226	C₂×C₃×C₄	—	4	4	no	—
227	C₂×C₃×C₄	—	4	4	no	—
228	C₂×C₃×C₄	—	18	1	no	—
229	C₂×C₃×C₄	—	18	1	no	—
230	C₂×C₃×C₄	—	18	1	no	—
231	C₂×C₃×C₄	C₃×D₁₂	1	4	yes	—
232	C₂×C₃×C₄	C₄⋊(C₃⋊D₃)	1	1	yes	characteristic
233	C₂×C₃×C₄	C₂²×C₃×D₃	1	4	yes	—
234	C₂×C₃×C₄	C₂²×C₃⋊D₃	1	1	yes	characteristic
235	#67	—	12	8	no	—
236	C₂×Dic₃	—	6	4	no	—
237	C₂×Dic₃	—	6	4	no	—
238	C₂×Dic₃	—	6	4	no	—
239	D₁₂	—	6	4	no	—
240	D₁₂	—	6	4	no	—
241	C₃×C₈	—	2	4	no	—
242	C₃×C₈	—	2	4	no	—
243	C₃×C₈	—	2	1	no	—
244	C₃×C₈	—	2	1	no	—
245	C₃×C₈	—	4	4	no	—
246	C₃×C₈	—	4	4	no	—
247	C₂×C₃²	—	2	4	no	—
248	C₂×C₃²	—	2	4	no	—
249	C₂×C₃²	—	2	4	no	—
250	C₂×C₃²	—	4	8	no	—
251	C₂×C₃²	—	4	4	no	—
252	C₂×C₃²	—	4	1	no	—
253	C₂×C₃²	#886	1	4	yes	—
254	C₂×C₃²	#904	1	4	yes	—
255	C₂×C₃²	#908	1	4	yes	—
256	C₂×C₃²	C₃×#123	1	1	yes	characteristic
257	C₂×C₃²	C₃×#135	1	1	yes	characteristic
258	C₂×C₃²	C₃×AGL(1,ℤ/8)	1	1	yes	characteristic
259	C₃⋊D₃	—	8	1	no	—
260	C₃×D₃	—	24	4	no	—
261	C₂×D₄	—	9	1	no	—
262	C₂×D₄	—	18	1	no	—
263	C₂²×C₄	C₂×C₃×C₃⋊D₃	1	1	yes	characteristic
264	C₂×C₈	C₂×C₃×C₃⋊D₃	1	1	yes	characteristic
265	C₂×C₈	C₂×C₃×C₃⋊D₃	1	1	yes	characteristic
266	C₄⋊C₄	—	9	1	no	—
267	C₄⋊C₄	—	9	1	no	—
268	C₄⋊C₄	—	9	1	no	—
269	C₂²⋊C₄	—	18	1	no	—
270	C₂²⋊C₄	—	18	1	no	—
271	C₄²	—	18	1	no	—
272	C₂²×C₃	—	2	8	no	—
273	C₂²×C₃	—	2	4	no	—
274	C₂²×C₃	—	2	4	no	—
275	C₂²×C₃	—	2	4	no	—
276	C₂²×C₃	—	2	1	no	—
277	C₂²×C₃	—	2	1	no	—
278	C₂²×C₃	—	2	1	no	—
279	C₂²×C₃	—	2	4	no	—
280	C₂²×C₃	—	4	4	no	—
281	C₂²×C₃	—	4	4	no	—
282	C₂²×C₃	—	36	1	no	—
283	C₂²×C₃	—	36	1	no	—
284	C₂²×C₃	—	36	1	no	—
285	C₂²×C₃	C₂×C₃×D₁₂	1	4	yes	—
286	C₂²×C₃	C₂×C₄⋊(C₃⋊D₃)	1	1	yes	characteristic
287	C₂×D₃	—	12	4	no	—
288	C₂×D₃	—	12	4	no	—
289	C₂×D₃	—	12	4	no	—
290	C₃×C₄	—	2	4	no	—
291	C₃×C₄	—	2	1	no	—
292	C₃×C₄	—	2	4	no	—
293	C₃×C₄	—	2	4	no	—
294	C₃×C₄	—	4	4	no	—
295	C₃×C₄	—	36	1	no	—
296	C₃×C₄	—	36	1	no	—
297	C₃×C₄	—	36	1	no	—
298	C₃×C₄	C₃×#237	1	4	yes	—
299	C₃×C₄	C₃×#237	1	4	yes	—
300	C₃×C₄	#3830	1	1	yes	characteristic
301	C₃×C₄	#3830	1	1	yes	characteristic
302	Dic₃	—	12	4	no	—
303	Dic₃	—	12	4	no	—
304	Dic₃	—	12	4	no	—
305	C₃²	—	2	4	no	—
306	C₃²	#4737	1	4	yes	complemented
307	C₃²	C₃×#482	1	1	yes	characteristic, complemented
308	C₂³	—	18	1	no	—
309	C₂³	C₃×C₄⋊(C₃⋊D₃)	1	1	yes	characteristic
310	D₄	—	18	1	no	—
311	D₄	—	18	1	no	—
312	D₄	—	36	2	no	—
313	C₂×C₄	—	2	1	no	—
314	C₂×C₄	—	2	1	no	—
315	C₂×C₄	—	18	1	no	—
316	C₂×C₄	—	18	1	no	—
317	C₂×C₄	—	18	1	no	—
318	C₂×C₄	C₃×C₄⋊(C₃⋊D₃)	1	1	yes	characteristic
319	C₂×C₄	C₂²×C₃×C₃⋊D₃	1	1	yes	characteristic
320	C₈	—	2	1	no	—
321	C₈	—	2	1	no	—
322	C₂×C₃	—	2	4	no	—
323	C₂×C₃	—	2	4	no	—
324	C₂×C₃	—	2	4	no	—
325	C₂×C₃	—	4	8	no	—
326	C₂×C₃	—	4	4	no	—
327	C₂×C₃	—	4	1	no	—
328	C₂×C₃	—	72	1	no	—
329	C₂×C₃	C₃×#886	1	4	yes	—
330	C₂×C₃	C₃×#904	1	4	yes	—
331	C₂×C₃	C₃×#908	1	4	yes	—
332	C₂×C₃	#64095	1	1	yes	characteristic
333	C₂×C₃	#64113	1	1	yes	characteristic
334	C₂×C₃	#64117	1	1	yes	characteristic
335	D₃	—	24	4	no	—
336	C₂²	—	2	1	no	—
337	C₂²	—	2	1	no	—
338	C₂²	—	2	1	no	—
339	C₂²	—	36	1	no	—
340	C₂²	—	36	1	no	—
341	C₂²	—	36	1	no	—
342	C₂²	C₂×C₃×C₄⋊(C₃⋊D₃)	1	1	yes	characteristic
343	C₄	—	2	1	no	—
344	C₄	—	36	1	no	—
345	C₄	—	36	1	no	—
346	C₄	—	36	1	no	—
347	C₄	C₃×#3830	1	1	yes	characteristic
348	C₄	C₃×#3830	1	1	yes	characteristic
349	C₃	—	2	4	no	—
350	C₃	C₃×#4737	1	4	yes	complemented
351	C₃	#10592436	1	1	yes	characteristic, direct product factor
352	C₂	—	4	1	no	—
353	C₂	—	72	1	no	—
354	C₂	C₃×#64095	1	1	yes	characteristic
355	C₂	C₃×#64113	1	1	yes	characteristic
356	C₂	C₃×#64117	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₃	C₂×C₄⋊(C₃⋊D₃)
commutator, cosocle	C₂×C₃²×C₄	C₂³×C₃
Frattini	C₂×C₄	C₂²×C₃×C₃⋊D₃
Fitting	C₃³×C₂²⋊C₈	C₂
socle	C₂²×C₃³	C₂×D₄
prime radical	C₃×#64139	C₂
hypercenter	C₃×C₂²⋊C₈	C₃⋊D₃
hypocenter	C₃²	C₃×#482
LCS	G ▷ C₂×C₃²×C₄ ▷ C₂×C₃² ▷ C₃²
UCS	C₁ ◁ C₂²×C₃ ◁ C₂²×C₃×C₄ ◁ C₃×C₂²⋊C₈

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

N	H ≅ G/N	type
2. C₃×#64135	C₂	semidirect
3. C₃³×C₂²⋊C₈	C₂	semidirect
4. C₃×#63648	C₂	semidirect
5. C₃×#63648	C₂	semidirect
6. C₃×#63645	C₂	semidirect
7. C₃×#63644	C₂	semidirect
8. #10592436	C₃	direct (N × H)
22. #64135	C₂×C₃	semidirect
26. C₃²×C₂²⋊C₈	D₃	semidirect
27. C₃²×C₂²⋊C₈	C₂×C₃	semidirect
28. #63648	C₂×C₃	semidirect
29. #63648	C₂×C₃	semidirect
30. #63645	C₂×C₃	semidirect
31. #63644	C₂×C₃	semidirect
99. C₃×C₂²⋊C₈	C₃×D₃	semidirect
100. C₃×C₂²⋊C₈	C₃⋊D₃	semidirect
210. C₂²⋊C₈	C₃×C₃⋊D₃	semidirect
211. C₃³	#482	semidirect
306. C₃²	#4737	semidirect
307. C₃²	C₃×#482	semidirect
350. C₃	C₃×#4737	semidirect
351. C₃	#10592436	direct (N × H)

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroup	index
C₃×#64139	2
C₃×#64135	2
C₃³×C₂²⋊C₈	2
C₃×#63648	2
C₃×#63645	2
C₃×#63644	2
#10592436	3
C₃×#4737	3

Subgroups — C3×#10592436 #13162774

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃×#10592436 #13162774